Enumeration problems for classes of self-similar graphs

We describe a general construction principle for a class of self-similar graphs. For various enumeration problems, we show that this construction leads to polynomial systems of recurrences and provide methods to solve these recurrences asymptotically. This is shown for different examples involving classical self-similar graphs such as the Sierpiński graphs. The enumeration problems we investigate include counting independent subsets, matchings and connected subsets.